Thesis (Ph.D.) - Cairo University - Faculty of Science - Department of Mathematics

This thesis is a contribution to numerical treatments for novel biological models of differential equations and their optimal control.These models are the fractional nonlinear Tuberculosis with multi-strain model, the hybrid fractional-order Coronavirus (2019-nCov) model, the fractional Tuberculosis infection model including the impact of Diabetes and resistant strains, the fractional Cancer treatment based on a synergy between anti-angiogenic and immune cell therapies model, the fractional-order model of Malaria, the fractional Tumor model under immune suppression. Some of these models are introduced here as fractional order with time delay models, these models are the fractional delay model of Tuberculosis with multi-strain and the fractional Cancer treatment based on the synergy between anti-angiogenic and immune cell therapies model. Also, the hybrid fractional-order stochastic Coronavirus (2019-nCov) model is studied. The concepts of the fractional derivatives in this thesis are Caputo, Grünwald- Letnikov, Riemann-Liouville fractional derivatives. Moreover, we used some new definitions for fractional derivatives such as Atangana-Baleanu-Caputo and hybrid fractional derivatives. General formulations for the fractional optimal control problems are introduced. A kind of Pontryagin{u2019}s maximum principle in fractional order case is used and the necessary optimality conditions are extended to the hybrid fractionalorder derivative. Moreover, we introduced in this thesis numerical treatments for two chaotic systems.These systems are the hybrid fractional-order finance system and the hybrid fractional Bloch system with time delay. Finally, we introduced numerical treatments for some novel partial differential equation models such as the variable-order fractional Klein Gordon equation, the two-dimensional linear and nonlinear fractional Cable equation, and the complex order fractional Burgers{u2019} equations

Issued also as CD